**SMS scnews item created by Daniel Daners at Wed 24 Jul 2013 1430**

Type: Seminar

Distribution: World

Expiry: 5 Aug 2013

**Calendar1: 5 Aug 2013 1400-1500**

Auth: daners@como.maths.usyd.edu.au

### PDE Seminar

# Perturbations of Complex Polynomials: Regularity of the Roots

### Parusinski

Adam Parusinski

University of Nice (France)

Mon 5 August 2013 2-3pm, Carslaw 829 (AGR)

## Abstract

We study the regularity of roots of complex polynomials \(P(t)
(Z)=Z^n+\sum _{j=1}^n a_j(t) Z^{n-j}\) depending on a real parameter
t. We first give an overview of the known results including in
particular the hyperbolic case (Rellich's and Bronshtein's Theorems).

Then we show, in the general case, that if the coefficients a_j(t) are
sufficiently regular (\(C^k\) for \(k=k(n)\) large) then any
continuous choice of roots is locally absolutely continuous. This
solves a problem that was open for more then a decade and implies that
some systems of pseudodifferential equations are solvable. Our main
tool is the resolution of singularities.
particular the hyperbolic case (Rellich's and Bronshtein's Theorems).

This is joint work with Armin Rainer from Vienna.

Check also the PDE Seminar
page. Enquiries to Florica Cîrstea or Daniel Daners.